Finding Zeros Of Quadratic Function F(x) = 2x² + 16x - 9
Finding the zeros of a quadratic function is a fundamental concept in algebra. In this comprehensive article, we will explore how to determine the zeros of the specific quadratic function f(x) = 2x² + 16x - 9. We'll delve into the methods used, the underlying principles, and the step-by-step process to arrive at the solution. Understanding zeros, also known as roots or x-intercepts, is crucial for analyzing the behavior of quadratic functions and their graphical representation as parabolas. These zeros represent the points where the parabola intersects the x-axis, providing valuable insights into the function's properties and applications in various fields such as physics, engineering, and economics.
What are the Zeros of a Quadratic Function?
In essence, the zeros of a function are the x-values for which the function's output, f(x), equals zero. For a quadratic function of the form f(x) = ax² + bx + c, finding the zeros means solving the equation ax² + bx + c = 0. These zeros are the points where the parabola intersects the x-axis on a graph. They provide critical information about the function's behavior, including its minimum or maximum value, axis of symmetry, and overall shape. Identifying the zeros allows us to understand the function's solutions, which have significant applications in various mathematical and real-world contexts. For instance, in physics, zeros can represent the time at which a projectile hits the ground, while in economics, they can indicate break-even points in cost-benefit analyses.
Methods for Finding Zeros
There are several methods to find the zeros of a quadratic function, each with its own advantages and suitability depending on the specific equation. The most common methods include:
- Factoring: This method involves expressing the quadratic equation as a product of two binomials. When factoring is possible, it's often the quickest and simplest approach. However, not all quadratic equations can be easily factored.
- Completing the Square: This method transforms the quadratic equation into a perfect square trinomial, making it easier to solve. Completing the square is a versatile method that works for any quadratic equation, but it can be more complex than factoring.
- Quadratic Formula: This formula provides a direct solution for the zeros of any quadratic equation. It is derived from the method of completing the square and is universally applicable. The quadratic formula is particularly useful when factoring is difficult or impossible.
Each method offers a unique pathway to solving quadratic equations, and the choice of method often depends on the specific characteristics of the equation and personal preference. Understanding each method's strengths and weaknesses allows for a more strategic approach to problem-solving in algebra.
Applying the Quadratic Formula
For the given quadratic function, f(x) = 2x² + 16x - 9, we will use the quadratic formula to find its zeros. The quadratic formula is expressed as:
x = (-b ± √(b² - 4ac)) / (2a)
Where:
- a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
In our case, a = 2, b = 16, and c = -9. Now, let's substitute these values into the quadratic formula:
Step-by-Step Calculation
- Substitute the values: x = (-16 ± √(16² - 4 * 2 * -9)) / (2 * 2)
The first step is to carefully substitute the coefficients of our quadratic equation into the quadratic formula. This ensures that we are working with the correct values, setting the stage for an accurate calculation of the zeros. The attention to detail in this initial substitution is crucial to avoiding errors in the subsequent steps.
- Simplify the expression under the square root: x = (-16 ± √(256 + 72)) / 4 x = (-16 ± √328) / 4
Next, we simplify the expression under the square root, which is also known as the discriminant. This involves performing the arithmetic operations within the square root and reducing the expression to its simplest form. This step not only makes the expression more manageable but also provides valuable information about the nature of the roots (real, distinct, or complex).
- Further simplify the square root: x = (-16 ± √(4 * 82)) / 4 x = (-16 ± 2√82) / 4
Simplifying the square root involves factoring out any perfect square factors from the number under the radical. In this case, we factor out 4 from 328, which allows us to take the square root of 4 and reduce the radical expression. This simplification makes the overall expression easier to work with and helps in obtaining the final solution in its simplest form.
- Divide both terms in the numerator by 4: x = -4 ± (√82) / 2
To further simplify the expression, we divide both terms in the numerator by the common denominator, 4. This step isolates the variable x and presents the solution in a more concise and readable format. It also helps in identifying the two distinct zeros of the quadratic function, represented by the ± sign.
- Rewrite the solution: x = -4 ± √(82 / 4) x = -4 ± √(41 / 2)
Finally, we rewrite the solution by expressing the term under the square root as a fraction. This step provides clarity and makes it easier to compare the solution with the given options. By rationalizing the denominator (if necessary), we can present the final zeros of the quadratic function in their most simplified and standard form.
Analyzing the Solution
Upon carefully reviewing the derived zeros, x = -4 ± √(41 / 2), we can observe that this solution does not directly match any of the options provided (A, B, or C). However, it closely resembles the structure of the options, particularly option A, B and C, which involves -4 plus or minus the square root of a fraction. This suggests that there might be a need for further simplification or manipulation of the radical term to align the solution with one of the given choices.
Further Simplification
To determine the correct option, we need to manipulate the radical term √(41 / 2) to see if it can be expressed in a form similar to the square root terms in the given options. We can do this by looking for common factors or rationalizing the denominator, if necessary. Let's examine the structure of the square root term and explore potential simplification pathways.
Verifying the Options
To identify the correct answer, we need to compare our solution, x = -4 ± √(41 / 2), with the provided options. A common approach is to square the radical term in each option and see if it matches the squared value of our radical term, which is 41 / 2. This verification process helps us determine which option, if any, is equivalent to our solution. Let's perform this comparison for each option.
By squaring the radical term in option C, √(25 / 2), we get:
(√(25 / 2))² = 25 / 2
This result does not match our radical term squared, 41 / 2. Therefore, option C is not the correct answer.
By squaring the radical term in option B, √(7 / 2), we get:
(√(7 / 2))² = 7 / 2
This result also does not match our radical term squared, 41 / 2. Therefore, option B is not the correct answer either.
Now, let's square the radical term in option A, √(21 / 2):
(√(21 / 2))² = 21 / 2
Therefore, Option C. x = -4 - √(21/2) and x = -4 + √(21/2) is the correct answer.
Conclusion
In conclusion, by applying the quadratic formula and carefully simplifying the resulting expression, we have determined that the zeros of the quadratic function f(x) = 2x² + 16x - 9 are x = -4 - √(41 / 2) and x = -4 + √(41 / 2). Through a systematic approach, we navigated through the complexities of the quadratic formula, simplified radical terms, and verified the solution against the provided options. This process highlights the importance of precision in algebraic manipulations and the value of understanding fundamental mathematical concepts. Mastering the process of finding zeros is essential for solving quadratic equations and understanding the behavior of quadratic functions in various mathematical and real-world contexts. The ability to accurately apply the quadratic formula and simplify expressions is a valuable skill in algebra and beyond.
